Question

Assume that the risk-free rate, Rf = 5%; the expected rate of return on the market, E(Rm)= 11%; and that the standard deviation of returns on the market portfolio is σM =20%. Calculate the expected return and standard deviation of returns for portfolios that are 25%, 75%, and 125% invested in the market portfolio.

Answer 1

	Please note following Points
	1) Standard Deviation of Risk Free Asset (σRf) is 0.
	2) For Risk Free Asset, as Standard Deviation is 0,
	Covariance = Correlation with any other risky asset
	will be 0.
	Therefore,
	Standard deviation of a Portfolio
	= √W1^2*σM^2 + W2^2*σRf^2 + 2*r*W1*W2*σM*σRf
	W1 = Weight of Market
	σM = Standard Deviation of Market
	W2 = Weight of Risk Free Asset
	σRf = Standard Deviation of Risk Free Asset = 0
	r = Correlation between Market and Risk Free Asset = 0
	So,
	Standard deviation of a Portfolio
	= √W1^2*σM^2 + W2^2*σRf^2 + 2*r*W1*W2*σM*σRf
	= √W1^2*σM^2 + W2^2*0^2 + 2*0*W1*W2*σM*σRf
	= √W1^2*σM^2 + 0 + 0
	= √W1^2*σM^2
a)	25% invested in Market Portfolio
	Expected Return of Portfolio
	= Weight of Market*Return of Market + Weight of Risk Free Asset*Return of Risk Free Asset
	= Weight of Market*Return of Market + (1-Weight of Market)*Return of Risk Free Asset
	= 25%*11% + (1-25%)*5%
	= 2.75% + 0.75*5%
	= 2.75% + 3.75%
	= 6.50%
	Standard Deviation of Portfolio
	= √W1^2*σM^2
	Where,
	W1 = Weight of Market = 25% = 0.25
	σM = Standard Deviation of Market = 20% = 0.20
	So,
	Standard Deviation of Portfolio
	= √W1^2*σM^2
	= √0.25^2*0.20^2
	= √0.0625*0.04
	= √0.0025
	= 0.05
	i.e. 5%
b)	75% invested in Market Portfolio
	Expected Return of Portfolio
	= Weight of Market*Return of Market + Weight of Risk Free Asset*Return of Risk Free Asset
	= Weight of Market*Return of Market + (1-Weight of Market)*Return of Risk Free Asset
	= 75%*11% + (1-75%)*5%
	= 8.25% + 0.25*5%
	= 8.25% + 1.25%
	= 9.50%
	Standard Deviation of Portfolio
	= √W1^2*σM^2
	Where,
	W1 = Weight of Market = 75% = 0.75
	σM = Standard Deviation of Market = 20% = 0.20
	So,
	Standard Deviation of Portfolio
	= √W1^2*σM^2
	= √0.75^2*0.20^2
	= √0.5625*0.04
	= √0.0225
	= 0.15
	i.e. 15%
c)	125% invested in Market Portfolio
	Expected Return of Portfolio
	= Weight of Market*Return of Market + Weight of Risk Free Asset*Return of Risk Free Asset
	= Weight of Market*Return of Market + (1-Weight of Market)*Return of Risk Free Asset
	= 125%*11% + (1-125%)*5%
	= 13.75% + (-0.25)*5%
	= 13.75% - 1.25%
	= 12.50%
	Standard Deviation of Portfolio
	= √W1^2*σM^2
	Where,
	W1 = Weight of Market = 125% = 1.25
	σM = Standard Deviation of Market = 20% = 0.20
	So,
	Standard Deviation of Portfolio
	= √W1^2*σM^2
	= √1.25^2*0.20^2
	= √1.5625*0.04
	= √0.0625
	= 0.25
	i.e. 25%